## Question about subset

Hello, I was looking into this proof

http://www.proofwiki.org/wiki/Lipsch...lly_Equivalent

and I was wondering how they concluded that

$$N_{h\epsilon}(f(x);d_2) \subseteq N_{\epsilon}(x;d_1)$$
$$N_{\frac{\epsilon}{k}}(f(x);d_1) \subseteq N_{\epsilon}(x;d_2)$$

Couldn't it also be that

$$N_{h\epsilon}(f(x);d_2) \supseteq N_{\epsilon}(x;d_1)$$
$$N_{\frac{\epsilon}{k}}(f(x);d_1) \supseteq N_{\epsilon}(x;d_2)$$

Thanks!
 Blog Entries: 8 Recognitions: Gold Member Science Advisor Staff Emeritus You have proven that if $y\in N_{h\varepsilon}(f(x);d_2)$, then $y\in N_\varepsilon(x;d_1)$. This implies that $N_{h\varepsilon}(f(x);d_2)\subseteq N_\varepsilon(x;d_1)$. Indeed, saying that $A\subseteq B$ means exactly that all $y\in A$ also have $y\in B$.
 Thanks ;)
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